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On finite shear

par Boulanger, Philippe ;Hayes, Michael
Référence Archive for rational mechanics and analysis, 151, 2, page (125-185)
Publication Publié, 2000-03

Article révisé par les pairs

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Résumé :

If a pair of material line elements, passing through a typical particle P in a body, subtend an angle Θ before deformation, and Θ + γ after deformation, the pair of material elements is said to be sheared by the amount γ. Here all pairs of material elements at P are considered for arbitrary deformations. Two main problems are addressed and solved. The first is the determination of all pairs of material line elements at P which are unsheared. The second is the determination of that pair of material line elements at P which suffers the maximum shear. All unsheared pairs of material elements in a given plane π(S) with normal S passing through P are considered. Provided π(S) is not a plane of central circular section of the C-ellipsoid at P (where C is the right Cauchy-Green strain tensor), it is seen that corresponding to any material element in π(S) there is, in general, one companion material element in π(S) such that the element and its companion are unsheared. There are, however, two elements in π(S) which have no companions. We call their corresponding directions limiting directions. Equally inclined to the direction of least stretch in the plane π(S), the limiting directions play a central role. It is seen that, in a given plane π(S), the pair of material line elements which suffer the maximum shear lie along the limiting directions in π(S). If ΘL is the acute angle subtended by the limiting directions in π(S) before deformation, then this angle is sheared into its supplement π - ΘL so that the maximum shear γ* in π(S) is γ* = π - 2ΘL. If S is given and C is known, then ΘL may be determined immediately. Its calculation does not involve knowing the eigenvectors or eigenvalues of C. When all possible planes through P are considered, it is seen that the global maximum shear γ*G occurs for material elements lying along the limiting directions in the plane spanned by the eigenvectors of C corresponding to the greatest principal stretch (λ3) and the least (λ1). The limiting directions in this principal plane of C subtend the angle 2 tan-1 ((λ1/λ3)1/2) and γ*G = π - 4 tan-1 ((λ1/λ3)1/2).